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13   TOPOLOGICAL VECTOR SPACES       59

(Observe that, if s is close to e in G, then the image of H under the mapping
xisxs"1x~1 is close to e.)

8. Let G be a locally compact metrizable group and H a closed normal subgroup of G,
such that G/H is not discrete and has no small subgroups. Show that there exists a
continuous homoniorphism/: R ~^G such that the composition R->G->G/H is
nontrivial (see Section 12.9, Problem 7).

13. TOPOLOGICAL VECTOR SPACES

We shall adhere to the conventions of (5.1), so that all vector spaces under
consideration have as field of scalars either the real field R or the complex
field C.

If E is a vector space over R (resp. C), a topology on E is said to be com-
patible with the vector space structure if the mappings (x, y)i~+x + y of
E x E into E, and (1, x) h~> fa of R x E (resp. C x E) into E are continuous. A
vector space endowed with a topology compatible with its vector space
structure is called a topological vector space. Since  x = (I)*, the condi-
tions above imply that the topology is a fortiori compatible with the additive
group structure of E, and all the notions and properties established for
topological groups in the preceding sections will apply in particular to topolo-
gical vector spaces (provided of course that these properties are transcribed
into additive notation).

An isomorphism of a topological vector space E onto a topological vector
space F is a linear bijection of E onto F which is a homeomorphism.

If || A: || is a norm on a vector space E, the topology on E defined by the
distance \\x  y\\ is compatible with the vector space structure of E (5.1.5) and
therefore makes E a topological vector space. Two equivalent norms (5.6)
define the same topology.

In a topological vector space E, the translations x\~*x + a and the
homotheties x\-+fa(A ^ 0) are homeomorphisms of E onto E, the inverse
mappings being, respectively, the translation x\~-+x  a and the homothety

In a vector space E over R (resp, C) a set M is said to be balanced if,
for each x e M and each scalar A such that |A| ^ 1, we have Ax e M (in other
words, if AM c M whenever \A\  1). The set M is said to be absorbing if, for
each x e E, there exists a real number a > 0 such that /be e M whenever
|/l| <jl a. Every absorbing subset of E generates the vector space E. For a
balanced subset M of E to be absorbing, it is enough that for each x e E there
should exist a scalar A ^ 0 such that fa e M.(b)   Deduce from (a) that if HI, H2 are two normal subgroups of a topological group